The determinant | Chapter 6, Essence of linear algebra

[@yujinpark6374]

This man deserves a Nobel Teaching Prize

[@minhtamtran7857]

He does.

[@sushilkumarlohani6709]

So true

[@piotrczekaa364]

But there is no Nobel Prize for maths, because Nobel hated mathematicians

[@lapofontana6220]

Gay

[@Tanvir_Ahmed_Earth]

Professor Dave is better 😊

[@alecyates3767]

"Understanding what it represents is, trust me, much more important than the computation" Said none of my courses involving determinants over the past decade, and why years later I am still looking up this stuff on youtube! This channel is amazing.

[@donlansdonlans3363]

Same in my algebra class, the professor just defined the determinant, stated a few properties without proof and gave us a worksheet that had like 50 exercises. All of them asked to compute the determinant of a matrix

[@alecyates3767]

@Fluffybrute Ofcourse you also need to learn how to apply the concept, but if you don't have any conceptual understanding, you forget why it was being used in the first place. Years later, you won't even recognise when a problem might call for taking the determinant. Ultimately, you can revisit the computation once you've identified the value in doing so. Judging from some other comments I have seen, it is clear you are very involved in mathematical subjects, so this viewpoint might be more difficult to understand for you. I am speaking for the people who have not visited this subject area for many years, so it is not fresh in the memory. In my current work, it would be useful if I could recall "oh, I think this kind of problem is related to X, let me check how to apply this". Unfortunately, years of only focussing on computation and not conceptual understanding has rendered me unable to do this. I feel like mathematics is one of the few subjects (based on how it is often taught) where you can go through a University degree, get an okay grade, yet feel, years later, that you know nothing about the subject. That is rare compared to other degrees.

[@DD-vc7fq]

I am a 5th year math student and I didn't know all of this about determinant. Kinda sad, isn't it?

[@lerui2820]

It's so bizarre to expect someone to feel motivated to know what these kinds of stuff are, when all you are given is: find determinant, knowing determinant a, and b being a linear transformation of a, find determinant of b. No one gives context

[@lerui2820]

@@alecyates3767 That last segment summarizes people I know talking about college.

[@lukitas009]

I have a master's degree in mechanical engineering and I'm starting to think I should redo my whole education from ground up searching for this kind of intuitive knowledge. It's absurd that I find out explanations which are as intuitive as this one so late in my life. I'm blown away completely! I mean how many bits of information have we stumbled upon during our formal education failing to see how they elegantly relate to each other and form a bigger picture...oh my!

[@gamerxdking4369]

i feel the same way and i have a masters degree in mathematics

[@noxfelis5333]

Our educational systems completely fails to teach us a lot of very important skills, showing us the bigger picture and the importance of some small detail. I don't know how many years you have left in the field, but rereviewing atleast a few things is probably worth it, even if it is just for the sole purpose of satisfying your curiosity and as a side bonus it will most likely make you a better engineer.

[@nayjer2576]

I am in second semester in pure math and I work hard to get these intuitive knowledge, thinking about one definition for 2 days sometimes. I don't know, I get nothing out of my lectures, only books, my own thought processes and such beautiful videos get me something.

[@noxfelis5333]

@@nayjer2576 I remember my professors just rewriting pretty much what was in the book already when doing pure math, it wasn't very enlightening as it can be difficulty to have a general grasp of the bigger picture of how it works and when to use it with a proof. My recommendation would be to read on the stuff pre-lecture in a way where you focus on the concept and what they are trying to achieve first before you dive in deeply into the proofs and calculations. For example, line integration (also refered as curve integration) is done on a vector field, so eatch point in space will have its own vector and it is expressed within the integral itself, whereas the line you are integrating on is the trajectory of something that moves throught that vector field.

[@nayjer2576]

@@noxfelis5333 That's a good idea, thank you. I try to catch up right now with the lectures, I am a little bit behind. But I will do that if I catched up.

[@spiralhalo]

As a programmer, knowing what these represent is astronomically more important than how to compute them for a good reason. Thank you.

[@patrickmayer9218]

Sad that most Linear Algebra courses are taught from an engineering/get-the-right-answer approach rather than a pure math/understanding-what's-going-on approach.

[@Laevatei1nn]

can you tell me why? im currently on the first semester and already dying trying to understand these

[@patrickmayer9218]

@@Laevatei1nn I don't know the applications for other sectors of CS, but the determinant is important for game dev because the area can be used for things like Area of Effect physics.

[@shenawy04]

​@@patrickmayer9218 Pretty important for machine learning, too

[@ArmageddonPhysics]

@@patrickmayer9218Tf you talking about? Engineers do everything except get the right answer, sometimes going to great lengths of laziness to do so intentionally

[@treyforest2466]

6:13 How can you say “parallelepiped” is the best shape name ever when mere seconds ago you were using the far superior “slanty-slanty cube”?

[@wisn6327]

petition to switch to slanty-slanty cube

[@themasstermwahahahah]

Slanty slanty cube would be on the level of how snap crackle pop somehow became official names

[@navry01]

Ze Frank would be proudy-proud

[@arulpraveen]

Exactly the same opinion after hearing it :)

[@richardschaffer5588]

Squished most rigorously!

[@kavishkabartlett4261]

You have absolutely no idea how much your videos have made me appreciate linear algebra. I always understand the how and why, but never what everything actually represented. Don't have much to spare because I'm a broke college student lmao but your content is just helpful it wouldn't be fair to just take it for free. Hopefully everyone who watches donates at least a little so you can keep doing what you're doing!!

[@joshuadutton5046]

lmao hey kavishka

[@cvcvv4]

What a beautiful idea

[@cbhorxo]

He'd be a millionaire if that were the case

[@_strangelet__]

​@@cbhorxo At least he deserves it, unlike most "influencers" who ain't adding anything to the society.

[@badlydrawnturtle8484]

That's what determinant is? Seriously? Why don't they just say that in the textbook? I spent days of my life wrestling with the idea that they wanted me to compute a magical number using an arbitrary formula.

[@dianac4480]

Badly Drawn Turtle i know it's so fucking frustrating

[@michaelbauers8800]

I am with you. And I got a minor in math in college. I mostly understood diff EQ, and linear algebra, and vector analysis...but I was confused WTF a determinant meant.

[@madelinescyphers5413]

Michael Bauers I have a degree in math and only learned this today while studying for my GRE. Like, why is this not taught in every single linear algebra class?

[@michaelbauers8800]

@@madelinescyphers5413 They don't want us to know! :)

[@matt2.019]

Because people are less competent than they would have you believe. This is why questioning authority is so important.

[@Haz2288]

DO. NOT. STOP. MAKING. THESE.

[@AnkitGupta-du1wf]

Stop using . as a delimiter

[@AlchemistOfNirnroot]

@@AnkitGupta-du1wf Shhhhhhhhh

[@IQstrategy]

@@AnkitGupta-du1wf Cheek & tongue - period indicates stop or end. Ur right naughty but funny ;)

[@indoorusedonly]

@@AnkitGupta-du1wf why you watch, you passed long time ago no?

[@elimiNator345]

I think you need more dot period things

[@_strangelet__]

If the matrix M1 scales any area "A" to "cA", and M2 scales any area "A" to "dA", so this means that det(M1) = c and det(M2) = d, which implies det(M1)det(M2) = cd. Now, if we consider the matrix M1M2, it is essentially like scaling the area "A" first by matrix M2, and then by matrix M1. So, when we first transform "A" with M2, the area becomes "dA". Then, when we transform this new area "dA" with matrix M1, we know that M1 scales any area by a factor "c", so the new area becomes "cdA", hence we can conclude det(M1M2) = cd. This shows that det(M1M2) = det(M1)det(M2).

[@switchwithSagar]

"Then, when we transform this new area "cA" with matrix M2" -> Don't you think M2 should be replaced with M1 in your comment. M2 scales the area by 'c' Hence the new area is cA. Now comes the M1 transformation which scales 'cA' to (d)cA.

[@_strangelet__]

@@switchwithSagar Yes! actually. Thanks for pointing it out. A silly mistake on my end.

[@emilsinkovic692]

that's more than one sentence

[@_strangelet__]

@@emilsinkovic692 Umm..I can remove all punctuation marks and make it one...if you want

[@leosammy5257]

@@_strangelet__ Well i would suppose since matrix multiplication is associtative, then getting the distinct determinant of M1 and multiplying with that of M2 is equal to solving the resultant matrix of M1M2 and finding its determinant.

[@phi111ie]

These videos are truly amazing. Thank you so much for making them.

[@johnduffy2777]

My barber is named Brandon

[@Llooktook]

totally agree, my uni professor was awful at explaining this and suddenly I am beginning to understand it intuitively

[@sad_man_no_talent]

Money

[@uripatootie7064]

damn a $100 donation, what an absolute giga chad

[@Aetherakabane]

​@@johnduffy2777so what?

[@camkiranratna]

This is what I thought about the property Grant mentioned in the end. Multiplying two matrices means that we are applying one transformation , then the other. The first transformation scales a unit area by “c” , and the second transformation scales the scaled area by “d”. So the overall scaling for the 1x1 unit square is “c” times “d “ . Now, looking at the right hand side we have the product of determinants. Since the determinants of the respective matrices are “c” and “d” , their product is “c times d”. If anyone has a better explanation please let me know. Thank you for your time .

[@charlesz88]

I think your explanation makes the most sense to me. Thanks!

[@kakalimukherjee3297]

Excellent man excellent. This is what maths is about. Intuition

[@rishabbomma9361]

Man more or less that’s what I thought to

[@yeetholmes619]

Thanks mate

[@yingfan3350]

Hi. This is absolutely a nice explanation. But more specifically, I'd like to say based on the left hand side, the transformation is scaling a unit area by 'c' (for M2) first and then by 'd' (for M1), while the right hand side do the scaling by 'd' (for M1) first and 'c' (for M2). The determinant is scalar, so the order doesn't matter. Based on this rational, we could conclude det(M1M2) = det(M2M1) as both of them equal to det(M1)det(M2). Actually this conclution is pretty interesting, as we know M1M2 doesn't equal to M2M1, but the determinants of these two are equal.

[@slowpoke7785]

5 years after my first lecture on determinants I finally understand it’s nature

[@KC-wm6tr]

same here, i wish i had this much earlier so that i could get an A in the examination

[@TheProblembaer2]

Same here, that's why I watch his brilliant videos. To actually understand this.

[@jiannisDimi]

Yes, it seems that we are mostly victims in the hands of lunatics who name themselves "mathematicians", and they may be indeed, but they are not teachers, they dont have the ability to interpret abstract ideas and make students visualise the simple thruth about them.... Unfortunately the majority of zhe teachers in schools have studied a topic, but they are not able to teach it...

[@marcfinns4302]

35 years...

[@saptashwapaul5583]

I hope this video helped you evolve, into a slowbro ;)

[@UKXcalibur]

i almost started crying cause i finally understand what a determinant is. THANK YOU

[@volodymyrhavrylov7993]

Same here.

[@dansantner]

same

[@алексейфедоровичкарамазов]

For how long did you cry?

[@idrisShiningTimes]

@@алексейфедоровичкарамазов lol

[@thiccalbert]

@@алексейфедоровичкарамазов Let's take a determinant to figure that out.

[@sasamilic720]

You've achieved the impossible ... demystifying the determinant

[@T33K3SS3LCH3N]

And it turns out the truth is actually really simple to understand, farmoreso than the cryptic explanations typical math teachers give! What a surprise.

[@TheNefastor]

He determined the determinant. Inception ensues.

[@MrSonny6155]

Now then, onwards to monads.

[@amir_os754]

Kinda? I think that in some sense he wasn't actually teaching the determinant. I think its more like he created a model for what the determinant is. The way math teachers in uni seem to teach it is as if its some proven (rigorous), abstract, truth. When you apply it to something spatial it makes it easier to understand intuitively, but it will still be only a model of the abstract, algebraic mathematics. (E.g: You can think intuitively understand that det(M1M2) = det(M1) * det(M2), but you need to prove it with math to be sure that the model that you're thinking of actually describes the underlying reality.) So in some sense the determinant is still mystified, but we have some intuition about it. It sort of reminds me of Plato's forms.

[@BlackTablewood]

@@amir_os754 That is kind of a 'thing' in math. The more you understand, the more you are confronted with magic and unicorns. And dragons. And demons. Unfortunately fewer women though, but the drama happens in different dimensions.

[@alejrandom6592]

Your videos have really changed my way of seeing mathematics. It's sad that the school system has grabbed something beautiful, cut off all the intuition, turned it into a chore, and just makes us memorize formulas instead of actually understanding the logic behind them. Math isn't about the numbers, it's about knowing how to go back and forth between the visual and the abstract. You sir, are my hero.

[@physicshuman9808]

Yes and when Mathematics is the use of the intellect of the soul to measure His creation and the flow of it. Indeed numbers are a proof of Allah. One wants to bring the intuition of it to show the beauty of it 👍

[@Piffsnow]

I'm a math teacher and I didn't even know all of that. Why nobody told us in uni ? These videos are great, really, but they would have been more useful to me 9 years ago. :/ Anyway, thanks a lot !

[@zairaner1489]

You probably were taught this, but the results were just very very hidden in other (more general) results. And ofc, specifically the view at determinants like it is shown here is normally (at least in my uni) not proved in Linear Algebra, bit in Analysis/Measure theory

[@Piffsnow]

I genuinely don't remember the fact that the determinant tells you how much the area of the unit square changes... But I may have forgotten, obviously.

[@unev]

I've had linear algebra a year ago and there was nothing like this. I assume such kind of representation of a material is still, unfortunately, an exception and not the rule. Most people around me have this idea in their heads 'just pass the exam', and its seems so few are concerned with developing a complete mental model of the subject which enables you to come up with your own receipies.

[@XmatigX]

Well, when you're not taught any of the deeper meaning, linear algebra becomes a rote course where you kind of just learn the steps, and get good at identities and tricks. I wish they had taught it to me like this. I've always struggled with linear algebra, as I am an intuitive learner, and never had anyone teach it to me in a way that could be intuited.

[@XmatigX]

None of what he showed was nature. It was all apriori geometric arguments. All perfectly fine for intuition. Also, I think you're wrong. The most important application of math is to use it to describe nature. Why not understand it by looking at nature in the first place? Sometimes the best way to solve a puzzle is to work from both sides. If we have the answer, why not work backwards?

[@jacquesnicolay9221]

I'm a 4th year Mechanical Engineering student and have had algebra classes, calculus, trig, Engineering analysis, linear algebra etc... I just learned that a determinant is the factor by which a transformation is scaled lmao

[@TheMazinka]

Jacques Nicolay I’m a third year student and after all those classes, I finally also get to learn that the determinant is just a scaling factor 😂

[@zherox434]

@@TheMazinka wow I feel lucky to know this before going to university lol

[@ash_ithape]

@@zherox434 From this we learn that do not solely rely on the education delivered in uni . Self study is a bliss :)

[@seanwool]

Welcome to the MEWDKWADWUN Club - Mechanical Engineers Who Didn't Know What a Determinant Was Until Now

[@francisosullivan2558]

I'm a 4th year computer engineer joining the club haha. Doing research on machine learning is demanding knowledge on matrices that I lost about 2 years ago. Wish it was explained like this to me before, would have made my matrices class so much easier.

[@smortwonbon7267]

This video is so helpful, my uni never told us what a determinant actually is, we were just expected to compute it. This is really making me appreciate math a whole lot more and is motivating me to study harder. Thanks a lot!

[@MrRoflHamster]

Damn. That's actually interesting. At University the determinent was just that number. 'Here, go compute that, it's important,'

[@jaxsharp]

Why not?

[@huh5950]

It might sound stupid but I nearly cried seeing this because for the first time since I started uni this year or even since I started middle school I feel like im deeply understanding the basic concepts and not just banging my head against the book trying ro get it in my head by memorizing, thank you from the buttom of my heart

[@anukalpabhaumik5157]

Me too. I feel so grateful. My mind feels so light and stable like it is ready to understand more and not stuck or lost while new concepts keep piling up. These videos finally gave some meaning and sense to what I have been doing in uni and why and what it all meant.❤

[@AryanSajith]

9:30 The space scaled by M2 then M1 is equivalent to the space scaled by the linear composition of said two matrices(since linear composition combines the two linear transformations).

[@noname-ue3oh]

I'm confused doesn't that imply commutivity of M2 and M1? As det(M2)*det(M1) is commutative If so ,why so? If not, why not?

[@disabledbee487]

​​@@noname-ue3oh i believe you're mixing up the matrix properties with scalar properties, multiplication is of course commutative and when the computing the determinant the final answer is of scalar value and not a matrix hence with multiplication of Det(M2)*Det(M1) = Det(M1)*Det(M2) because of the determinant being a scalar. Of course as you know this doesn't apply to matrix multiplication, meaning Det(M1*M2) ≠ Det(M2*M1)

[@gatewaytofarts2771]

@@disabledbee487 Det(M1*M2) = Det(M2*M1) is true determinants are commutative but matrix multiplication is not. its because 2 matries can have the same determinate so even tho M1*M2 ≠ M2*M1 their determinates are the same. Geometrically the detriminates are the same because M1*M2 and M2*M1 have the same shape but they are rotated differently so they are represented by different matries but have the same area. At least I think I know that algebraically the determinates are the same but Im not 100 sure if they have the same shape but are rotated. Sorry for spelling errors

[@girishshivnauth9507]

it scaled by the same factor for both instances to the determinant does not change@@noname-ue3oh

[@RealAjGhimire]

continuing on with the intuition, I am thinking of M1 as I-hat and M2 as j-hat. I think that makes things fall in place neatly just like other example..

[@aniruddhasanyal7625]

I am really glad I found this series and channel while still in high school.

[@ShauriePvs]

lucky!

[@pmvslakshmi7397]

same here dude we are in the same boat

[@aldyfernando4570]

i found this channel when im just graduate -_-"

[@Fuzzyhead5060]

That’s so lucky man ! I just wrote an exam on this yesterday and I wish I had of watched this two days sooner !

[@guacamol.e]

lucky man! I've found this just as I'm trying to teach the concept to my brother in high school. I really wish I'd found this in high school or college.

[@Titurel]

i was more excited waiting for this video than for The Avengers...

[@gnanay8555]

Well, that's not really surprising, is it ?

[@hawkeye2958]

yes

[@zes3813]

wr

[@tabacclastname2336]

Awesome Now i have started to love maths

[@Tekkerue]

3Blue1Brown...SMASH! ;)

[@devatraijha4952]

The reason why I love Khan Academy and 3B1B is because they make learning feel like natural intuition and not forced memorisation. The first helps me (and a lot of students i suppose) get deep into a concept, from introduction to numericals, while the latter brings the underlying geometry and visualisation to life. Shout out to the brilliant educators!😀

[@amrapalisamanta5085]

True

[@CJIANG]

This fantastic video definitely deserves more attentions. The only thing I can do is taking a few seconds to write this short comment to show my gratitude.

[@3blue1brown]

I'm glad you liked it. This might be my own favorite in the series so far.

[@tangdexian3323]

better, become a patron

[@cybisz2883]

Woah. I completed a mathematics degree, but no professor of mine ever related determinants with area/volume. In my Linear Algebra course, the text book just gave us the definition of a determinant (i.e., how to compute them), then proceeded to discuss their properties via abstract theorems and proofs (e.g., how determinants of different matrices are related, cramer's rule, how they show whether a matrix has an inverse or not), but there was absolutely nothing really intuitive about them. But that was only one unit of the course. We moved on to vector spaces and other topics without ever really using determinants again except when we needed to know whether matrices were singular (well, except for a brief excursion into eigenvalues/vectors that we had to rush through due to time constraints) I realize mathematics is a giant field, but as I explore topics on my own part of me feels a bit cheated. There were numerous topics that were taught clumsily at my university that, now that I know them better, should have been easy for my professors or text books to explain clearly. It's so unfortunate. Thank you for your videos, 3Blue1Brown. They're done exceptionally well and have definitely helped my understanding.

[@zairaner1489]

Didn't you had multidimensional integrals in your Analysis courses?

[@cybisz2883]

+Raphael Schmidpeter There was a lower-division 3 semester calculus sequence that all science/engineering majors took, the third of which included a great deal on multidimensional integrals. The closest thing to determinants in that course, however, was heavy application of cross products. But again, it was taught as just an algorithm for getting a vector that's perpendicular to two others - no real explanation of why the cross product does that. The upper division Real Analysis course that math majors took didn't get into multivariable calculus - but it basically reconstructed single-variable calculus rigorously over arbitrary metric spaces from the perspective of set theory (off the top of my head, lots on the different forms of continuity, connectedness & compactness, the different ways of defining integration, proving Taylor's theorem and Heine-Borel theorem, and many other related topics).

[@zairaner1489]

Cybis Z You may or may not know, but for functions in n-dimensionsal space, the derivative is replaced by a derivative matrix, and the determinant of that matrix in s point x tells you how much a small volume around x is changed. The strongesg version of that Statement is the multidimensional Substitution rule which is like the one dimensional, but the derivative is replaced by the determinant of the derivative matrix. (The Statement of the video is an easy corollary from that)

[@cybisz2883]

+Raphael Schmidpeter That's pretty cool. I definitely did not know that. I thought that for n-dimensional space, the derivative is replaced by either the gradient vector, or the normal vector to the plane tangent to the function at the given point. I don't quite understand this "derivative matrix" though - are you referring to a Jacobian matrix? I thought that would only be square if you're working with a whole vector of functions, not just a scalar function in n-dimensional space.

[@zairaner1489]

Yes exactly I mean the Jacobi matrix, should have used that word. And of course we are talking about functions from n-dimensional space to n-dimensional space (otherwise there would be no sense about talking how a volume gets cahnged considering everything gets sqished into 1d)

[@arunvasudevan3961]

3rd year in aerospace engineering currently. Having to touch up on linear algebra for an upcoming midterm (TOMORROW!). No teacher is able to intuitively explain these concepts at a high-level, so since freshman year, I’ve been intimidated by linear algebra in general. It is a crime that people think it’s okay to basically instruct students to plug and chug without understanding anything. People like you are making the next generation of STEM students confident and capable. Thank you!

[@EugeneKhutoryansky]

Good explanation.

[@muqtarjamaegal6071]

But yours better

[@dhairyaparam]

what is this... a crossover episode ?

[@muqtarjamaegal6071]

What are you mean

[@abrownbearification]

You are my hero

[@FireJayceFTW]

omg the best crossover you will ever see

[@blablabla12a]

4 years after seeing this video and finishing my EE degree, I can safely say this is one of the best videos I've seen that helps grasp the true meaning of such a basic concept in linear algebra. Every time I compute a determinant my brain recalls the visualizations in this video and it helps me understand what I'm actually doing. Thank you Grant.

[@shairuno]

it takes a lot of works and understanding to simplify the abstract idea into a simple visual understanding - great job!

[@syakirhisyam9779]

"If you scale the sides of any rectangle twice, its area is same as if you are multiplying the areas of rectangles formed by individual scaling." M1M2 transforms space by M2 then by M1, scaling it by the scale factor of M2 then by the scale factor of M1.

[@brimussy]

ah, thankyou!!

[@syakirhisyam9779]

@@brimussy hahaha nice

[@frankdimeglio8216]

@@brimussy WHAT IS E=MC2 is taken directly from F=ma, AS TIME is NECESSARILY possible/potential AND actual ON/IN BALANCE; AS ELECTROMAGNETISM/energy is CLEARLY AND NECESSARILY proven to be gravity (ON/IN BALANCE); AS the rotation of WHAT IS THE MOON matches the revolution. Consider TIME AND time dilation ON BALANCE. The stars AND PLANETS are POINTS in the night sky ON BALANCE. The diameter of WHAT IS THE MOON is about one quarter of that of what is THE EARTH. On balance, the density of what is the Sun is believed to be about one quarter of that of what is THE EARTH. Excellent. Consider what is THE EYE ON BALANCE. The TRANSLUCENT AND BLUE sky is CLEARLY (and fully) consistent WITH what is E=MC2. WHAT IS THE EARTH/ground is fully consistent WITH what is E=MC2. CLEAR water comes from what is THE EYE ON BALANCE. Notice what is the fully illuminated (AND setting/WHITE) MOON AND what is the orange (AND setting) Sun. They are the SAME SIZE as what is THE EYE ON BALANCE. Lava IS orange, AND it is even blood red. Yellow is the hottest color of lava. The hottest flame color is blue. What is E=MC2 is dimensionally consistent. WHAT IS E=MC2 is consistent with TIME AND what is gravity. What is gravity is, ON BALANCE, an INTERACTION that cannot be shielded or blocked. Consider what are the tides. The human body has about the same density as water. Lava is about three times as dense as water. The bulk density of WHAT IS THE MOON IS comparable to that of (volcanic) basaltic lavas on what is THE EARTH/ground. Pure water is half as dense as packed sand/wet packed sand. Now, the gravitational force of WHAT IS THE SUN upon WHAT IS THE MOON is about twice that of THE EARTH. Accordingly, ON BALANCE, the crust of the far side of what is the Moon is about twice as thick as the crust of the near side of what is the Moon. The maria (lunar “seas”) occupy one third of the visible near side of what is the Moon. The surface gravity of the Moon is about one sixth of that of what is THE EARTH/ground. The lunar surface is chiefly composed of pumice. The land surface area of what is the Earth is 29 percent. This is exactly between (ON BALANCE) one third AND one quarter. Finally, notice that the density of what is the Sun is believed to be about one quarter of that of what is THE EARTH. One half times one third is one sixth. One fourth times two thirds is one sixth. By Frank Martin DiMeglio

[@dantingzeng5281]

I'm a little confused. for example if we have a 2*2 square, and scale by 2 and 5, wouldn't the result be 4*4 and 10*10, and 20*20, which makes 1600 and 400?

[@dantingzeng5281]

I have a better explanation. The answer is not multiplying the areas of rectangles but instead multiplying the scale. 2*2 -- 4 4*4=16 is 4 times larger than 4 10*10=100 is 25 times larger than 4 20*20 is 100 times larger than 4 4*25 is 100, which means the scaling is conserved.

[@chandus2496]

I have a bachelors in mathematics and I find these concepts STILL new! I don't understand why or how I have never come across such intuitive explanations of what these operations does! I have lost count of the number of books on LA I have read and none of them bothered to mention what is probably the most important intuition behind what a determinant is?

[@aeroscience9834]

Chandu S would you mind listing these books you’ve lost count of? I’d be interested in reading them

[@uzairakram899]

@@aeroscience9834 Have a read through: Linear Algebra and its Application by David C Lay Fifth Edition. The pdf can be found online free. Chapter 3 covers determinants. Report back if you find any explanation of intuition behind determinants.

[@Deksudo]

9:37 applying the two transformations consecutively has the effect of multiplying the original area by those determinants, and applying the composed transformation matrix is essentially the same thing as consecutive application of the composed transformations so it'd only make sense for the composed matrix's determinant to be the same as the separate determinants' product.

[@tristanshah9716]

You are a better teacher than any I've had to pay for in college.

[@HighlyShifty]

@@dsu1 What does Judaism or Russia have to do with anything?

[@kjekelle96]

0:00 intro 0:22 stretching space 2:28 the "determinant" 3:42 negative determinants 5:30 in 3D 7:32 how to compute? 9:18 peanut butter jelly time

[@natopal0819]

I've been taking higher mathematics classes for about 6 years, have learned about vectors/matrices in quantum mechanics, maths and physics and i still DIDN'T even know what the value of a determinant really represents.... today i am thankful 4 you 3B1B, because this made me understand sooooo many things i'd just learned and accepted without even knowing why they follow certain rules. THANK YOU

[@zeyneperguven4985]

I thought I made the comment... I am studying physics for 5 years and I just felt that bro... We always tried to calculate what the determinant is, but what was the f*ckin determinant actually? The video is amazing, loved it! Thank you!

[@Jee2024IIT]

@@zeyneperguven4985 I am a high school student and i thought you were supposed to get told about the actual meaning of determinants in uni

[@tristanmoller9498]

@@Jee2024IIT That really begs the question: What path did 3B1B take to actually learn this? I’m in engineering (bachelors) currently and this is also my first time learning what the determinant is. Seems like, it’s like that for most everyone watching. And, who are the crazy smart people to come up with all of these concepts? We are using plenty of linear transformations in my mechanics: dynamics class.

[@arslanahmadbhatti6267]

I rather use one word. Associativity

[@swarnimbarapatre5353]

Is it normal to cry by getting overwhelmed while watching these videos?

[@NovaWarrior77]

Let it out, it's all good.

[@GammaFZ]

RTX Off: Let it out, it's all good. RTX On: Permit its evacuation, the emotions are all positive.

[@elliottandreasen4148]

Swarnim Barapatre well depends on how old you are ;)

[@jellybabiesarecool4657]

It's fine as long as you then remember to pause and ponder to find where you're stuck.

[@muhammadridho7680]

@@GammaFZ shhhhh

[@helo3827]

I ask my teacher: What is the determinant? My teacher: The determinant is the determinant!!!

[@finmat95]

No shit Sherlock

[@keshavladha3108]

My textbook, definition "A determinant is a 2×2 square containing four numbers"😂

[@Shreyasiiiii]

I don't have enough courage to ask this question to my math teacher🤣

[@Shreyasiiiii]

@@keshavladha3108 perfection 🤣👌

[@miccujha8115]

@@keshavladha3108 😂😂😂😂

[@B0XMATTER]

You're exploding my brain with this. I'm dealing with some point cloud transformations, which are essentially just big 3-dimensional matrices, and it's ridiculous how complicated it looks if you only look at it on paper, but then it's visualized and explained to you, it suddenly becomes intuitive as hell. I think I'm going to add a very special acknowledgement in my master's thesis. Thank you ever so much.

[@T33K3SS3LCH3N]

What an eye opener. I had multiple teachers and professors tell us about the determinant and NONE of them were able to explain to us what it actually is, and why it's so meaningful that it appears in other formulas. It took you less than three minutes to do that.

[@philoposos]

If this channel existed +/-15 years ago, my life would have been sooo different...

[@nigelwang2447]

7 episodes into the Linear Algebra series and my biggest takeaway has been that I never truly understood Linear Algebra. It seemed so painful and non-sensical back then, but I truly appreciate the elegance and simplicity now.

[@JavaPythonsVids]

slanty-slanty cube is the best not-actually-a-real-term ever

[@henryg.8762]

YESS

[@algorythmis4805]

p a r a l l e l e p i p e d

[@siddharthjoshi6811]

People like you should be cherished. I am from India, and in old times, the teachers were given highest regard. And i think you deserve that regard ...

[@learninglearner]

facts bhai

[@funwithsiblings7310]

These videos... Wow, they're just gold... Such simple and concise visual explanation makes me wonder "that's it? Why are we not taught this?" Also, thank you so much for always leaving us with something to "pause and ponder"... That's where those lightbulb moments occurs and you're like wow, that's big brain... one of those moments where you ask a question and we feel proud to answer it, because you've taught us the concepts... Even if we don't answer, it ignites a curiousity You make us understand why things are what they are rather than just telling what they are... It's almost like I'm a mathematician who's laying down the foundation for years of curriculum to come.. the logic and not just the arbitrary rules Your videos have the power to give goosebumps Salute man!!

[@AvantGrade]

6:51Columns are linearly dependent because c3 is a linear combination of c1 and c2 ( c3 = c1 + c2 ).

[@lizardking640]

thank you, scrolled a lot to find it

[@ronak859]

Thanks man

[@vincent3542]

the simple term is because they all lay down in same line, which is linear dependent

[@duykhanh7746]

you can also say that c1 is a combination of c2 and c3 xD

[@bilsid]

@@vincent3542 or plane in a 3d space

[@soniczdawun1]

Coming off of the idea that each matrix represents a linear transformation of a space, the matrix product M1M2 would represent M2s transformation followed by M1s transformation, which streches space by a factor of det(M2), then again by a factor of det(m1), so the resulting net scaling coefficient would be the product det(M2)*det(M1), which is, because of the commutative property of scalar multiplication, equal to det(M1)*det(M2); at the same time, the matrix product M1M2 is a unique matrix that represents just one linear transformation, which only has one scaling factor of det(M1M2), making it equal to the split up scaling product before.

[@manideepsridhara2831]

Guy Edwards amazing

[@saichandra3116]

He is the only youtuber i have seen who gets likes,shares and subscriptions without even saying a word to do so in the video...... Great way to teach and this method should be adopted everywhere to make students understand in the schools Thanks 3B1B :)

[@Czimchik]

Why universities are so reluctant just to say it! It's like they never wanted us to know what we are doing! Amazing job, man. Thank you for all of this.

[@gordongoodwin6279]

Whales are better than cats

[@veeek8]

Agreed, university text books are often written in an inexcessible way

[@Mr.Nichan]

@@veeek8 Actually, my textbook explained it but my teacher didn't (or maybe he did actually, he did go over at least one geometric interpretations chapter once). (Actually, I often find that the textbooks contain a lot of the material that's more interesting and often also explanations that are more informative than what's in the lecture.) Maybe it's partially because it's hard for the teacher to draw such complicated things in lecture (though he did pre-write/draw a lot of his notes and other teachers use computer-drawn presentations in other classes I took that might make it easier to see). There's also a point my Dad made because he took a quantum physics class ("Quantum Theory of Matter" or something; I think they computed electron orbitals and such; he passed but learned nothing from it, btw., so he's not sure himself) from someone who actively disliked visualizations and talked about how they give you shallow understanding and misconceptions, which is that that my Dad wonders if there are some mathematicians/etc. who actually understand things better with numbers than with visuals somehow, and that they're teaching the way they would want to learn.

[@mylovelykids5795]

no words !!! i spent more than 2-3 days to search and understand this and felt so happy when i came across this.. yes as many said below, we learn maths as only formulas and numbers in the universities and just mug them for competing in the rat race.. feel shame to claim as maths toppers once which was achieved without knowing any essence of it !! PLEASE DO NOT STOP THIS.. PLEASE CONTINUE !!!!

[@ghostek7792]

im not taking linear algebra until january 2023, but I genuinely find so much interest in mathematics. I jsut got finished with calc 3 and I hear that it's a good class to take prior since it utilizes some similar concepts. It's going to be great learning while having an intuitive understanding of what it is im trying to accomplish before I even take the class. Thanks brother

[@xyzct]

Dear University, may I please have four years of tuition back? Thank you.

[@uzairakram899]

@@NomadUrpagi Not in the engineering department, In fact for my school the engineering building is 4 miles out of the main campus.

[@NomadUrpagi]

@@uzairakram899 hahaha lulw. Rip man

[@uzairakram899]

@@NomadUrpagi I would actually go out of my way to go to the main campus, its worth a shot but I would have to buy the $300 parking pass which is not enforced in the engineering building so I'm kinda stuck between a rock and a hard place.

[@NomadUrpagi]

@@uzairakram899 yeap it happens. Engr dept is always separated from main buildings. What about friday nights? You go out to town to drink like other students? Assuming this is not a conservative country

[@uzairakram899]

@@NomadUrpagi I'm in America, but I commute an hour every day, living with my parents and they are too conservative too allow going out to town to drink and get laid, and under their roof its their rules.

[@javierbg1995]

I love how this is making me think about matrices like I hadn't before. Here's something I just realised: We already know that A * B ≠ B * A (they are not necessarily equal) , since the order in which you apply transformations affects the resulting combined transformation. However, because det(A * B) = det(A) * det(B), that means that the scaling factor of the transformations is NOT affected by the order in which they are applied. That's so cool! Any thoughts on this?

[@3blue1brown]

Really good observation. It's a nice little shadow of commutativity in a non-commutative world.

[@abdelrhmanahmed1378]

lol

[@saqibnawaz10]

Its like (speed=5m/s) and (velocity = 5m/s north) are both by definition different but they have same magnitude in common. Idk if that makes any sense🤔

[@tedsioris3828]

Would that imply then, that det(A*B) = det(A) * det(B) = det(B*A)

[@monkeybusiness673]

I have just run one simple example with numbers and it turned out to be true. I have no clue how to formally prove that, however. But it seems intuitively right.

[@AdalineKafarne]

As determinant defines invertion of the full space and scaling factor of area in that space, it turns out that when applying Matrix multiplication, we can tell the total scaling factor and how much times space was inverted, and both don't rely on the order of linear transformations, so basically thats why the determinant of composition can be described with a multiplication of determinants of separate linear transformations parts of that composition

[@saequayasmeen7023]

Sal Khan and 3blue1brown. The perfect duo.

[@AkamiChannel]

Saequa Yasmeen I need my Walter Lewin

[@ShomeAvi]

thwnx for that name...I was searching cell cone or something lol

[@ssgsaxgamer4902]

I think 3blue1brown actually did the multivariable calculus series on khan academy

[@a.v.w.odavid6979]

haha i was watching this series yesterday around midnight and my dad suddenly burst into my room and though what all fathers would think in that situation. he thought i was watching porn and the look on his face when i told him i was watching "the essence of linear algebra episode 3" the look on his face was absolutely priceless

[@3blue1brown]

Well...I'm glad I could redirect your efforts.

[@fred321cba]

Perhaps you can tell your dad that this has taught you the importance of having your own space.

[@underworldling]

His dad would likely agree, but might want to linearly transform that space using a matrix with a small determinant.

[@sabouedcleek611]

LOL

[@playerguy2]

+Tyler Poole soo around 0? or get him out of that space by annoying hum with a negative one?

[@stuffingspace4912]

OH MY GOD THIS WAS SO HELPFUL…. Literally on my last week of linear algebra and I’ve been wondering what all this stuff actually means…. Everything makes so much sense now, your videos are such a life saver

[@victorarnault]

Oh my God... I'm on the ground now. How beautiful. I'm almost cry.

[@yahavx]

I*

[@just.a.guy522]

8:44 This... This is beautiful. This right here is beautiful. You can literally understand it in less than a minute and it's not mentioned ANYWHERE in the textbooks I've read. This is the answer to the question "WHY AD-CB?" that every student studying linear algebra has had. Truly beautiful, and helpful.

[@ZULFAZLIBINHAMJAH]

Seriously, this maybe the first ever a person able to explain determinant as humanly possible. No textbook explain as clearly as you explain

[@matthewsiu2397]

Every linear algebra course should require this series as a primer. Having this background makes things so much clearer.

[@aBigBadWolf]

If you can do this with probabilities as well, gosh I'd have to marry you or something.

[@pdcx]

lol

[@liuzhen2008]

OMG pls. Do it for the statistics stuff. Whats PDF? WHats a moment?? WHATs a moment generating function??

[@egor.okhterov]

And calculus.

[@seanki98]

+Охтеров Егор he has done mv calc ;)

[@aBigBadWolf]

link?

[@backhand4]

My linear algebra course was all computation and proofs. Proofs, with little-to-no visualizations, did not capture the essence of linear algebra. I took the course in 2006 and did not have videos like this at the time. The idea of getting to supplement my course material with resources like this today almost makes we want to go back to school! This is absolutely brilliant.

[@Yunas_Jet]

I took a fundamentals of linear algebra course last semester at University, and got a reasonable intuition on most of the course, but until I watched this video I could not for the life of me understand determinants, and so just computed all the raw numbers to get the right answer. One question I particularly struggled on one homework was the one that you pose at the very end of this video. A few months ago, it took me almost an entire evening to work through it, and I still didn't understand my proof. After watching these videos, within about 15 seconds I understood exactly *why* it was true. Sure, that understanding wouldn't hold water when the lecturer's asked you to rigorously prove it, but at least I'm finally starting to understand... well... everything I learnt in the entire last semester. Thank you so much for these videos, they're really helping, and I hope you make more after you're done with linear algebra! (And I'm doubly hoping that if you do they happen to be on differential equations since that's one of my topics next semester :P)

[@3blue1brown]

I'm really glad I could help. This is exactly the kind of story I was hoping to hear in making the series. As to the proof "holding water", I will say that once you prove that matrix multiplication actually does correspond to function composition and that the determinant really does scale areas, the intuition you have is indeed a completely rigorous proof.

[@galgrunfeld9954]

I liked a lot how you built intuition using geometric visualizaition, starting from 0 dimensions to 3 dimensions - all dimensions that we can experience visual (well, maybe not the point - but the idea of it). But I didn't like the fact that you didn't *show* generalizations of things so far, like determinants. I was able to do things in my head, in terms of intuition, like replacing 3-dimensional vectors with n-dimensional vectors. You also didn't do it in matrices - did a notation for n-ary matrix multiplication. It's like you stopped building the understanding at some point - you went up all the way to 3 dimensions, but then stopped. Why did you stop in just 3 dimensions, and took the intuition you built and used it to generalize examples you showed?

[@LeEnnyFace]

It does hold water because you can more easily back it with mathematical proof.

[@klyman1644]

det(MN) = det(M)det(N) means: Scaling factor by overall transformation MN = (Scaling factor by transformation M) x (Scaling factor by transformation N) Even shorter: Overall stretching equals first stretch then second stretch. Note: Same essence as chain rule for differentiation

[@mhm6421]

9:35 What I understand is first you transform a unit square etc. which has the area det(M2) because M2 is performed first. (Note that every term like square or area can be replaced by the dimension shapes of the given matrices). Then the M1 transformation occurs which should multiply the area you started with by det(M1) hence: det(M1 M2) = det(M1) det(M2)

[@abhinavpandey8456]

In One Sentence - "If you scale the sides of any rectangle twice, its area is same as if you are multiplying the areas of rectangles formed by individual scaling."

[@JatinSanghvi1]

This only capture the essence for two-dimensional shapes with no shear or rotation.

[@pendronator]

That's what I was thinking. The determinant is a scalar, so the intuition comes from scalar multiplication.

[@sandypan777]

@@pendronator Say one scales up the area and the next one scales down. The final result will be the same as multiplying area of them both.

[@SwavimanKumar]

@ASUH DUDE True. Area does change with shearing.

[@valeriobertoncello1809]

No, because "multiplying areas" gives you 4D measures.

[@nicholasholfester3685]

this channel makes me love math more and more every day

[@faustobarbuto]

One single minute of these videos is worth hours of classes and study. You're unveiling a new Linear Algebra world to me.

[@rishabhjain472]

I'm almost done with my masters in math and I feel like this is the first time I'm learning Linear algebra. Thanks a lot man !! God bless you

[@antoinegaillard9440]

Students now are so lucky to have this kind of content!

[@bmorr]

I'd say I'm pretty proficient at Linear Algebra right now within the class I'm taking at university, but it's as if my professor explains how to solve Linear Algebra problems, and 3b1b explains why Linear is the way it is. Which helps me understand it more intuitively. I love the information from this video because it makes so many of the applications of determinants make sense!

[@tedsheridan8725]

It took me years of tutoring / teaching linear algebra to develop this kind of intuition. I always try to show these concepts with pen and paper, but the animations are brilliant. Thanks!

[@MarkCidade]

Since matrix multiplication is a composition of functions, the determinant is distributive over matrix multiplication because you are just applying the scale of one parallelogram's area to the other's area, whether you multiply the matrices first or you multiply each of their determinants separately (the determinant function is a _homomorphism_ for invertible matrices under multiplication).

[@SGManiac1255]

more succinctly, the determinant from the first transformation, becomes the "unit square" of the second.

[@y__h]

+SGManiac1255 the unit square is in the form of first transformation of base vector?

[@SGManiac1255]

the transformed version of the original 1x1 unit square effectively becomes the new 1x1 unit square for the 2nd transformation. which means you multiply the new transformation's determinant by the first one.

[@bnouadam]

it is exactly this kind of hermetic argument for which a a video like this is necessary. det(M1*M2)= V2/Vo=(V1/V0)*(V2/V1) = det(M1)*det(M2). V0 the initial unit volume generate by the basis vectors, V1 the so called stretched or squished volume upon the first transformation V2 the volume after the second transformation but relative to the initial volume which is V1.

[@hamzazaidi7042]

SGManiac1255 Damn thanks for putting it this way

[@gokulraj3272]

Whenever I learn something new in school I come here to learn it again. This channel just flips my world upside down with the new insights into the concept.

[@leoads]

You are the man. Absolutely Awesome!!! I'm a strucutral engineer with 20 years of experience, and I always wanted to some what the hell means the determinant of a matrix. You just change my life! thank you!

[@anchitbhattacharya9125]

This stuff is life changing! Seeing your video, made me realize that it's all about imagination. Everything stems out of an imagination, and we use numbers and methods to communicate it with others. But at the end nobody teaches us how to imagine, but teaches us the results of the imagination, and the results without the imagination is meaningless!

[@vpr17]

9:40 On multiplying M2 the space is first scaled to det(M2) and basis' move accordingly, then multiplying by M1 scales the transformed area by a factor of det(M1). The resultant scaling on the space becoming det(M1) * det(M2).

[@raakshasan418]

isn't it that M1*M2 is just M1(M2) as in it's a composite operation?

[@harveyhensley875]

9:00 "Go watch Sal Khan work through a few" :-}

[@Alejandro-hh5ub]

This has been extremely enlightening. I would love to watch more series of videos like this featuring other topics within mathematics. A series on Calculus would be keenly appreciated.

[@3blue1brown]

I'll keep that in mind for the future.

[@foobargorch]

While you're taking requests.... ;-) randomness is another subject for which good intuitions are important, and I'm sure your visual method would really help. I often find myself trying to explain crypto things to people, and at uni I was tasked with teaching basic stats for paper reading workshop (8 students), I decided to frame it as how to rationally decide if a pair of dice is loaded from first principles, as a sort of arc covering basic probability, absolute fundamental descriptive statistics, trying to predict a distribution and then predict the chances of seeing that distribution, with the end goal of being able to give the students a doorway into understanding what a quoted statistic actually implies. And I failed spectacularly. Even though I think it was still a good approach to the subject, I just couldn't get their eyes to not glaze over. It's such a hard subject to explain, and one that really affects our lives so much more than we're comfortable believing, generally speaking. I am still not really over how little the students seemed to care... blergh.

[@luizmeier]

Groups and Rings theory would be great too.

[@msolec2000]

Oh, are quaternions too high level?

[@gnanay8555]

Yay group theory ! :D

[@vitsobi]

This is a motivation to complete all the series on this channel. Imagine spending over a decade to realise that Determinant has more to do with Area than just a number. Thank you and thank you.

[@jcfos6294]

Extraordinaire en français, extraordinary in English!!! Great Great Great!!! Wonderful explanation!!! You're the best in algebra Mathématiques in youtube, in internet all over the world. Very very thank you. God bless you

[@akhileshmachiraju1521]

Since the same matrix transformation can be done in one step or multiple steps, the property that multiplies the determinant of the end matrix after complete transformation is equal to the product of determents of all the transformed matrices at multiple stages.

[@harmonyOfEureka]

I studied abroad in a non-english speaking country (Japan) and all of these are of course taught in Japanese. As a non-native who just started learning Japanese 2 years prior to college, I had a huge gap in my knowledge on these fundamental principles throughout the first 2 year even though I still passed the course on linear algebra. If not for your intuitive way of teaching, I don't even know how I could get a solid understand like today. Thank you Grant and your team for an amazing work.

[@gaureesha9840]

The answer for the last question is that when we are doing M1*M2, we first apply transformation of M2 and then do M1, that is from Right to Left. So when we do M2 first, the area of original block changes by some factor x, that is x *A. Now when we apply M1, again the area changes by some factor y, so the area is y*(x*A). This is same as multiplying individual determinant values, that is det(M1) * det(M2). Is this correct?

[@gauravmanwani9148]

Gaureesh A But don't you think this equation is dimensionally off note?

[@eeee2097]

no. it's not one sentence.

[@SlingerDomb]

you help me a lot man. but anyway i wonder how can one even answer this in just exactly one sentence!?

[@vinitparekh3524]

I dont know if this is correct. What if you were to unsheer (m1) and then sheer (m2) the factor difference would be x=.5 and y=.5 causing the area if A was 1 to be .25 However, the area would still be 1 if we were to shear and unsheer. I may be wrong, can someone please refine my thinking. Thank you

[@devangkapoor4159]

Vinit Parekh no you are wrong. The det(sheer) and det(unsheer) both equals 1 not 0.5 . You can see Det(sheer) in previous videos and the unsheer one can be calculated by finding the inverse of the sheer matrix ( you can see what I am talking about in the next video of inverse matrices at 4:42)

[@Supware]

Dude how did you hone your intuition like this..? Is it completely self-taught.?

[@Vaaaaadim]

I think people in general hone their intuition when they deal with such things frequently and have it fresh in their minds to think about.

[@HT-rq5pi]

-

[@AndreRhineDavis]

I personally find that only months or years after I learn something at school/university, after lots of playing around with the concepts myself and lots of *experience* with them, do I *truly* gain an appreciation for what's *really* going on

[@azlastor]

I agree in everything, there's nothing special about people that "just see it" other than the time they have spent throughout their life thinking about stuff and not learning to do things mechanically the way they were told to do so. They discover early in their maths life this easier (and perhaps more importantly more beautiful, less tedious and more satisfying) way of performing academically and therefore seek to apply it to every subject they come across, this in turn trains the skill. As performing academically becomes harder, and subjects become harder to grasp intuitively. Students that didn't "hone their intuition" face a bigger barrier of entry, eventually reaching a point where mechanical learning is, for them, the optimal way to perform academically. Once this point is reached this students often won't want to and/or will have a hard time learning intuitively, because they'll be aware that the extra effort will probably result in reduced academic performance, at least in the medium/short term. They are mathematically speaking stuck in a local minimum of effort/performance. This is all obviously just from my personal experiences, but after having tried by different means throughout my life as a student to help others to think more intuitively, finding this resistance is definitely the most frustrating experience for me. The "you just see it and I just don't" mentality is a common reaction and you end up falling back to mechanical teaching sprinkled with some intuitive insights. When I've talked about this issue with closer people they often think I'm dunning kruger-ing all over this but I've given it a lot of thought and I genuinely believe I'm not. All that's really needed is a little change in education when we are still kids, when this entry barrier to intuitive thinking does not exists, and we could all overcome this. If you are an educator and you are trying to do this a BIG thank you to you.

[@azlastor]

I'm not, I majored in engineering. I do believe education is what I had a passion for and probably I still do. I only ever taught people about my age and as I described I found that they would need this "push" in their way of learning math way earlier in their life, which is most likely why I took a deeper interest in education; just seeing how much of a difference it could have made. With my post I just wanted to thank you and any other educator out there for doing this, and wanted to elaborate on the idea you presented that once you are aware of intuitive thinking you search for more and therefore you become good at it and "naturally good at maths". I guess It just surprised me to see this: "Students that are aware of intuitive reasoning search for more and often get called gifted / naturally good at maths". Aware is quite honestly the perfect word here, also there's so much truth in this simple phrase and it's not said or thought enough. If more people believed it to be true, specially students and educators, the struggle with maths that most students face would disappear; most people would come to love math and see it's beauty; not dread it. And being math and intuitive thinking such powerful tools it would have really meaningful implications in their lives.

[@evanvandaniker]

These videos are literally blowing my mind, combing things I've learned in Calc 3 and Linear Algebra and Information Theory to explain math and space in the most understandable way possible. I'm pissed that this isn't how everyone learns math because it's so beautiful and makes so much sense

[@porterchild5617]

Thank you so much Grant! You're doing in hours what my teachers couldn't do in months. It's a surreal experience to realize that. Very conflicting emotions of outrage and enlightenment.

[@LucasDanielSantoro]

I'm feeling the same! And I've even passed the tests without knowing all this!

[@scirium7199]

I had a very basic introduction to linear algebra in my last year of high school, and I remember the professor (which I'm pretty sure is an actual mathematician) told us that matrices are really only useful to solve systems of equations and that determinants don't have much meaning beyond that. As someone who loves math, wow could that not be further from the truth.

[@luisemillerin207]

Same here, I remember directly forgetting about determinants after our teacher told us how "meaningless" they were. 🤦🏼‍♀ These videos are gold!

[@debasismt]

it brought tears in my eyes. I struggled a lot to have a physical interpretation of these in my college days, a decade back. Most of my friends mugged the theory and I spent lots of time just to understand what the matrix multiplication is, which I couldn't do with limited resources that time. and of course my marks were not good. You guys are lucky to have these free of cost, any time anywhere...

[@leoma500]

So thankful that this channel exists and appreciate all your work

[@dwarfyman4899]

M1M2 transforms space by M2 then by M1, scaling it by the scale factor of M2 then by the scale factor of M1.

[@affirmativedave.1846]

Great!

[@tamimabdullah4638]

can you please describe why M1M2 transforms space by M2 first the by M1 ??

[@RealLifeKyurem]

Tamim Abdullah From the video, we learnt matrices are transformations.Transformations and functions are equivalent, so we can think of compositions of transformations as composition of functions: M₂ ≔ g(x) M₁ ≔ f(x) v⃗ ≔ x M₁M₂v⃗ is then f(g(x)). Which function do apply to x first? That will be g(x). That’s why you apply M₂ first.

[@tamimabdullah4638]

@@RealLifeKyurem thanks mate 🙏. Awesome answer.

[@emiliatonello]

@@RealLifeKyurem and what would it be the equivalent of the chain rule for M1M2?

[@rachitmittal448]

If a matrix A scales the area by a factor of x and If a matrix A scales the area by factor y then it can be deduced that matrix AB would scale it by factor of xy. Example Let take an area of 5. If we quadruple it and then half it, It becomes only double. (5*4)*0.5 = 5*(4*0.5).

[@khizarpasha6105]

I'm watching this prepping for my lin alg class after calc 3, and oh my god the Jacobian makes so much sense now. This is actually crazy, in calc 3 they just tell you the jacobian is a way to adjust for aeea when converting between uv and xy space and that's literally what it is wow

[@AdamTrudeau]

I gone to school to learn about matrices. Through the whole course I had no idea what the determinant was. I was only shown how to find it and use it. In 3 minutes of this video I now understand the determinant better than from my 5 month university course (which I paid way too much for). Seriously, knowing the "why" is soooooooo important.

[@AdamTrudeau]

Also now that I know this, maybe we should not call it "determinant" but something that actually means something to someone trying to learn this. The word determinant meant nothing to me, in school, aside from determining that it broke things when it was equal to zero.

[@jaspercoggins3590]

9:30 Matrices are a scaling of their basis vectors via transformation, so by applying both transformations you're essentially multiplying their scalings on those basis vectors to determine where they land which will then determine the scaling of their area "shape" and thus space, hence the determinant. It's kind of like applying two scale factors to a shape consecutively vs applying the product of those scale factors to the same shape, you will get the same area regardless.

[@dstri1]

This channel should be added in our university curriculums.

[@arianeparadis6439]

I am currently reviewing Linear Algebra 1 to prepare for an advanced class and these are crucial details which should have be thought in my class but were not. I had no idea what the determinant stood for but only that it was related to the area and volume of a parallelepipied. Thank you so much for the visual representation. It makes a lot of sense.

[@shayabramov8059]

In our class we learn almost nothing about the determinant. its like 3-4 weeks we learn about matrices and their relation to equation systems, fields, inverse matrices, Gauss elimination. Then 3-4 weeks about vector spaces, spans, basis. Then we finish the semester with transformations, dual spaces, kernal and image spaces (where we also learn a *little* bit about determinant, but the teacher basically just tells us how to calculate it and its done in 1 lesson).

[@micocatipon145]

I have been struggling in math way back elementary til college. Then yet I bumped i to this channel because I need to review for a qualifying exam in data science. Man you are so good at teaching that I can easily focus and understand what you are explaining. I never hated math but I am Not good at it. Thank you bring clear! I’m 38 years old and it’s not too late to re-learn.

[@palaniappaprasanth2490]

Thanks a lot brother!!! I have seen no one explain matrices and determinants like you :)